On uniform convergence of piecewise-linear solutions to minimal surface equation

Abstract

In the paper we consider piecewise-linear solutions of the minimal surface equation over a given triangulation of a polyhedral domain. It is shown that under certain conditions, the gradients of these functions are bounded as the maximal diameter of the triangles of the triangulation tends to zero. It is stressed that this property holds if the piecewise-linear function approximates the area of the graph of a smooth function with a required accuracy. An implication of the obtained properties is the uniform convergence of piecewise linear solutions to the exact solution of the minimal surface equation.